Smoluchowski Diffusion Equation Research Articles

Overdamped Brownian motion in periodic symmetric potentials

The dynamics of an overdamped Brownian particle in the field of a one-dimensional symmetric periodic potential U(x;α) have been studied by numerical solution of the Smoluchowski diffusion equation and the Langevin equation using the Brownian Dynamics method. The parameter α controls the shape and height of the potential barrier, which ranges from a sinusoidal spatial dependence for low barrier heights (α small) to a near delta-function appearance for barrier heights tending to infinity (α very large). Both the mean square displacement (MSD) dα(t), and the probability density n(x,t|x0), where x0 denotes the initial position, have been calculated. The MSD over a wide time domain has been obtained for a number of values of α. The exact asymptotic (t→∞) form of the diffusion coefficient has been exploited to obtain an accurate representation for dα(t) at long times. The function, dα(t) changes its form in the range α=8–10, with the appearance of a “plateau” which signals a transition in the particle’s Brownian dynamics from a weakly hindered (but continuous) mechanism to essentially jump diffusion. In the limit α→∞, each well of U(x;α) becomes similar to the classical square well (SW), which we have revisited as it provides a valuable limiting case for dα(t) at α≫1. An effective “attraction” of the probability density towards the SW walls is observed for off-center initial starting positions, and it is suggested that this could explain an observed change in the analytic form of the SW MSD, dsw(t), at long times. Two approximate analytic forms for dsw(t) at short times have been derived. The relaxation of the Brownian particle distribution n(x,t|x0) in the initial-well of U(x;α) has been studied.

The Smoluchowski diffusion equation for structured macromolecules near structured surfaces

Beginning with the molecular-based Fokker–Planck equation obtained previously [M. H. Peters, J. Chem. Phys. 110, 528 (1999); J. Stat. Phys. 94, 557 (1999)], the Smoluchowski diffusion equation is derived here to describe the spatial and orientational dynamics of molecularly structured macromolecules near molecularly structured surfaces. The formal scaling and perturbation methods employed allow the establishment of definite limits on the use of the Smoluchowski equation when surfaces are present. It is shown that the Smoluchowski equation reduces to that given previously [D. W. Condiff and J. S. Dahler, J. Chem. Phys. 44, 3988 (1966)] in the absence of external surfaces. A specific example application is given involving a spherical macromolecule with electrostatic charge segments near a planar surface with an arbitrary charge distribution. Finally, we show that the short-time solution to the Smoluchowski equation obtained here yields a Brownian dynamics method consistent with that given previously [E. Dickinson, S. A. Allison, and J. A. McCammon, J. Chem. Soc. Faraday Trans. 2 81, 591 (1985)] for orientable, interacting Brownian particles. This study has applications to problems involving site-specific adsorption of orientable, structured Brownian particles, such as association or adsorption of biological macromolecules to cellular surfaces and enzyme–substrate docking kinetics, to name a few.

Femtosecond fluorescence upconversion studies of barrierless bond twisting of auramine in solution

Femtosecond fluorescence upconversion studies have been performed for auramine (a diphenylmethane dye), dissolved in ethanol, as a function of temperature. It is found that the (sub)picosecond decay components in the fluorescence slow down as the temperature is lowered from 293 K to 173 K. From the observation of a residual fluorescence, with a viscosity-dependent lifetime of about 30 ps (or longer at higher viscosity), and transient absorption results it is concluded that the two-state sink function model [B. Bagchi, G. R. Fleming, and D. W. Oxtoby, J. Chem. Phys. 78, 7375 (1983)] does not apply in the case of auramine. Comparison of the auramine fluorescence kinetics in ethanol and decanol shows that diffusional twisting and not solvation is the main cause for the (sub)picosecond excited state relaxation. To explain the experimental results, adiabatic coupling between a locally excited emissive state (F) and a nonemissive excited state (D) is considered. Torsional diffusion motions of the phenyl groups in the auramine molecule are held responsible for the population relaxation along the adiabatic potential of the mixed state, S1 (comprised of the F and D states). Simulation of the excited state dynamics is feasible assuming a barrierless-shaped potential energy for S1 and applying the Smoluchowski diffusion equation. The temporal behavior of the auramine band emission was simulated for the temperature range 293 K >T>173 K, with the temperature, T, and the viscosity coefficient, η, being the only variable parameters. The simulated temporal behavior of the emission in the investigated temperature range is compatible with that obtained experimentally. The rotational diffusion coefficient for the auramine phenyl groups as extracted from the simulations is found to follow the Einstein–Stokes relation. From the numerical calculations the effective radius of the twisting phenyl groups is determined as 1.0 Å which compares well with the actual value of 1.2 Å.

Dynamics of macromolecules and nuclear magnetic relaxation: Application of mode‐coupling diffusion theory to DNA, proteins and their complexes

AbstractA mode‐coupling approach has been used to solve the Smoluchowski diffusion equation describing the correlation functions relevant in nuclear magnetic relaxation experiments on biological molecules like protein and DNA. A good reproduction of relaxation patterns for molecules that are fluctuating about three‐dimensional structures imposed by secondary motives has been found. Important differences can be explained with the poor statistics collected by numerical simulations that do not contain fluctuations contributing to rates in the ns time‐scale, therefore comparable with rotational rates.

Cooperative diffusion in colloidal mixtures

In this work we develop a general method to examine cooperative diffusion, i.e., the partial dynamic structure factors, in multicomponent colloidal mixtures of spherical particles. Using a multivariable projection operator formalism based on the many-body Smoluchowski diffusion equation, we derive an exact microscopic expression for the matrix of irreducible memory functions associated with the partial dynamic structure factors and the long-time cooperative diffusion coefficients of the system. Starting from this microscopic expression, we present a derivation of a self-consistent mode coupling scheme (MCS) for cooperative diffusion in colloidal mixtures. This scheme accounts not only for the direct particle interactions, but also for the far-field part of the solvent mediated hydrodynamic interactions. Combined with our recent work on the mode coupling theory of tracer–diffusion and linear viscoelasticity of colloidal suspensions, this MCS provides a unified method for calculating dynamic and low-shear rheological properties of multicomponent colloidal dispersions. The MCS can be used to study polydispersity and mixing effects in concentrated colloidal systems. Some applications related to cooperative diffusion, interdiffusion, and glass transition in binary mixtures are discussed to illustrate the method.

Linear viscoelasticity of colloidal mixtures

In this work we develop a unifying and general method for calculating linear viscoelastic properties of multicomponent colloidal mixtures of spherical particles. Using linear response theory based on the many-body Smoluchowski diffusion equation, we derive an exact expression for the zero shear rate shear relaxation function, together with a Green-Kubo formula for the static shear viscosity. From these results, we obtain an exact expression for the high frequency elastic shear modulus of colloidal mixtures. We present, in addition, the first derivation of a self-consistent mode coupling scheme for the linear viscoelasticity of concentrated colloidal mixtures. This scheme offers the opportunity for a unified description of linear viscoelasticity and diffusion mechanisms. It accounts further for polydispersity and mixing effects, and leads naturally to a diverging shear viscosity at a glass transition point. Various limiting cases are considered to assess the accuracy of the approach. It is shown to be a valuable method for evaluating the rheological properties of concentrated colloidal mixtures.

Tracer-diffusion in colloidal mixtures: A mode-coupling scheme with hydrodynamic interactions

In this work, we develop a general theoretical scheme to study tracer-diffusion in mixtures of interacting colloidal particles where the influence of solvent-mediated hydrodynamic interactions is also considered. Based on the many-body Smoluchowski diffusion equation, we derive in a first step an exact microscopic expression of the irreducible memory function (self-friction function) associated with the self-intermediate scattering function and with the mean squared displacement of a tagged particle. By applying a mode-coupling scheme (MCS) to the irreducible memory function, we obtain explicit expressions for the tracer-diffusion quantities in terms of partial static structure factors and hydrodynamic functions. The influence of hydrodynamic interactions (HI) is accounted for using a far-field expansion of the two-body hydrodynamic diffusivity tensors. For charge-stabilized colloids, this is a good approximation due to strong electrostatic repulsion between the particles. Various applications are discussed in order to illustrate the versatility of our mode-coupling scheme.

Simulations of Solvation Dynamics Using a Nonlinear Response Approach

Simulations of the time evolution of the probability distribution function of the solvation coordinate of a solute in liquid solution are presented. Assuming impulsive excitation of the ground-state population to the excited-state free energy surface, the time dependence of the probability distribution function of the solvation coordinate is obtained by solving the Smoluchowski diffusion equation. Emphasis is on the influence of anharmonicity in the expression for the free energy surfaces on the solvation dynamics and its implications for time-resolved fluorescence measurements.

Femtosecond solvation and charge transfer dynamics in liquid solution

Time-resolved fluorescence up-conversion experiments have been performed for a few fluorescent organic charge transfer molecules in liquid solution. Dynamic Stokes shifts have been measured, with a time resolution of 150 fs, for the probe molecules in alcoholic and ethereal solvents. The results reveal solvation dynamics determined by “inertial free streaming” motions of the solvent molecules in the solvent cages and rotational diffusion motions of the solvent molecules. Simulations of the temporal changes in the fluorescence spectra based on the Smoluchowski diffusion equation are also briefly discussed.

Continuous versus discrete description of the transport in a membrane system

The continuous (macroscopic) and the discrete (microscopic) description of the membrane transport are related to each other. Specifically, we construct the Green's functions for a diffusive-convective system with a membrane (which is treated as a partially permeable wall) in three cases: in the first one the Smoluchowski (diffusion) equation (SE) is used, in the second case one applies the birth-death (BDE) equation, and finally, we combine the continuous and discrete description in such a way that in the space outside the wall the SE acts, whereas the passing through the membrane is described in terms of BDE. To obtain the Green's functions in the above-mentioned cases, we introduced a new generalized method of images.

Picosecond photolysis of azo compounds in liquid alkanes: germinate recombination kinetics for polyatomic free radical pairs

Picosecond optical absorption transients for the photolysis products of azocumene and the cyclic azo compound 3,8-diphenyl-1,2-diaza-1-cyclooctene have been measured in a series of alkane solvents having different liquid viscosities. The transient intermediate produced from azocumene decays through a diffusion influenced process which is interpreted as secondary recombination of geminate free radical pairs. This assignment is based on the wavelength dependence of the transient absorption signal, the viscosity dependence of the decay kinetics and the complementary decay profiles seen for free and tethered radical pairs. The long time limit of the survival probability for geminate cumyl free radicals follows the reciprocal square root fo time decay predicted by the Smoluchowski diffusion equation.

Interaction between alkali-metal ions M+ in the deactivation process of quartz-embedded Al-M+ centers.

The influence of the long- and short-range interactions between alkali-metal-ion compensators on the deactivation mechanism of Al-${\mathit{M}}^{+}$ centers (M=Li or Na) in quartz crystal during electrodiffusion is determined on the basis of a perturbation treatment of the Smoluchowski diffusion equation. Application to the deactivation rate of Al-${\mathrm{Li}}^{+}$ in the presence of another ${\mathrm{Li}}^{+}$ ion shows that the asymmetry induced in the frequency jump of the first ion by the Coulombic interaction with the second appears to be a localized effect. When the second ion lies in adjacent wells of the Al-${\mathrm{Li}}^{+}$ center, as expected in the model of diffusion and compensation, it is shown that the first-ion diffusion can be enhanced by two orders of magnitude with respect to the single-ion diffusion model without compensation.

Mean-field theories for multidimensional diffusion

Self-consistent-field (SCF) methods are developed for solution of multidimensional diffusion problems. Time-dependent self-consistent-field (TDSCF) equations are derived for the Smoluchowski diffusion equation, and are applied to a two-dimensional barrier crossing problem. This is compared to both time-dependent and time-independent SCF approximations derived for the Schrödinger equation in imaginary time, which is obtained by transformation of the diffusion equation. Results for the model problem show that the TDSCF approximation for the original diffusion equation is accurate, efficient, and readily implementable in higher dimensions. Applications to diffusion problems in condensed media are noted.

Solvent dynamical effects in electron transfer: Numerical predictions of molecularity effects using the mean spherical approximation

The influence of solvent molecularity upon the adiabatic barrier-crossing frequency νn and barrier height ΔG*, for electron–exchange reactions involving only solvent reorganization is examined numerically on the basis of a mean spherical approximation (MSA) treatment of the reaction coordinate time-correlation function, Δ(t) [Refs. 3(c) and 7(c)] for Debye solvents. The calculated ΔG* values for a spatially isolated redox couple fall increasingly below the corresponding dielectric continuum quantity ΔG*con as the ratio of the solvent to reactant radii (rsol /rre ) increases. For the experimentally common circumstance (rsol /rre )≲2 and for zero and infinite-frequency dielectric constants ε0 and ε∞ over the ‘‘typical’’ ranges 20–100 and 1.75–2.5, respectively, the calculated ΔG* values are up to ∼30% below ΔG*con , the deviations tending to be larger for smaller ε0 and ε∞ values. Two kinetic models are utilized to extract νn values from Δ(t): Hynes’ approach derived from the generalized Langevin equation (GLE), and that based on the Smoluchowski diffusion equation. The νn values derived using both models decrease progressively below the corresponding dielectric continuum frequency factor νconn as (rsol /rre ) increases, the deviations between νn and νconn being greater for smaller ε∞ and/or larger ε0 . The magnitude of these deviations, however, is noticeably smaller for the Hynes GLE than the diffusion model. Thus for (rsol /rre )≲2 with the above dielectric parameters, νn is calculated to be up to ∼two and five fold smaller than νconn on the basis of the Hynes GLE and diffusion model, respectively. These differences can be understood in terms of the relative influences upon νn of the slower relaxation components arising from short-range solvation, responsible for the deviations from νconn , in comparison with the faster dynamics associated with more distant solvent molecules. Slightly larger deviations from the continuum predictions are obtained using the GLE approach in the presence of barrier-top curvature. At least for (rsol/rre)<2, the MSA-prescribed rate constants also do not deviate greatly from the corresponding continuum prediction, the corresponding decreases in νn and ΔG* being partly compensatory.

The diffusion equation for a classical mechanical system in an anharmonic potential

A classical unidimensional mechanical system subject to a friction force and white noise is considered. The Langevin equation is derived by separating the velocity into a position-dependent component plus a fluctuating component with zero mean. Thereby the Smoluchowski equation for diffusion with corrections up to ϕ(1/β 3) is obtained, and an approximate closed equation for the mean value of the velocity is established.

A Fokker–Planck equation for canonical non-Markovian systems: A local linearization approach

Correctly renormalized drift coefficients can be straightforwardly derived using the linear version of the generalized Langevin equation and linear reaction potentials (parabolas or inverted parabolas). Recent investigation via computer simulation of molecular dynamics and numerical solution of stochastic differential equations, shows that interesting cases exist where the nonlinear nature of the interaction between reacting system and ‘‘bath’’ and that of the reaction potential must be taken into account. Then it is shown that a Smoluchowski diffusion equation with correctly renormalized drift coefficients can be obtained by adopting a local linearization assumption, which, nevertheless allows the reaction coordinate to ‘‘feel’’ the influence of different transport properties in different regions of the reaction potential. Under the special condition where the system–bath interaction is assumed to be linear, this Smoluchowski equation is shown to coincide with that recently proposed by Okuyama and Oxtoby [J. Chem. Phys. 84, 5830 (1986)]. In the case where the renormalization corrections are neglected, this equation coincides with that proposed by our group [Fonseca, P. Grigolini, and D. Pareo, J. Chem. Phys. 83, 1039 (1985)].

Geminate recombination in proton-transfer reactions. III. Kinetics and equilibrium inside a finite sphere

The kinetics of reversible dissociation of a single ion pair inside a finite sphere is described by the transient solution of the Debye–Smoluchowski diffusion equation, with backreaction boundary conditions at contact, and a reflective boundary condition at the sphere’s radius. At long times the solution tends to an equilibrium state. A simple analytic expression for the equilibrium coefficient is derived. For an infinitely large sphere (‘‘infinite dilution’’) it reduces to an expression similar to that of Fuoss, while its dependence on the sphere’s radius is qualitatively similar to the concentration dependence of activity coefficients.

Theory of intramolecular spin relaxation by translational diffusion in locally ordered fluids

The theory of spin relaxation induced by translational diffusion of small molecules or ions in locally ordered fluids, developed in parts I and II of this series, is extended to cylindrical geometry as, for example, in polyelectrolyte solutions or in hexagonal lyotropic liquid crystals. The theory is based on the Smoluchowski diffusion equation with nonuniform potential of mean force and translational diffusivity and on the cylindrical cell model. Formally exact closed-form expressions are derived for the zero-frequency spectral density associated with radial diffusion, while a general numerical algorithm is described for computing the full frequency-dependent spectral density. Several useful approximations, such as the dynamic cell approximation, the steady state approximation and the surface diffusion approximation, are formulated and their accuracy quantitatively assessed. Calculations are reported for a mean-field interaction model based on the nonlinear PoissonBoltzmann equation, with emphasis on applications of the theory to counterion spin relaxation in polyelectrolyte solutions. l. INTRODUCTION In locally ordered fluids, such as macromolecular solutions and lyotropic liquid crystals, nuclear spin relaxation can be induced by modulation, through translational motion of the fluid molecules, of the magnitude and orientation of the residual intramolecular spin-lattice coupling, which is only partially averaged by local molecular motions near an interface. In part I I-1] and part II I-2] of this series of papers, we developed a theoretical framework, based on the so-called continuous diffusion model (CDM), for treating spin relaxation in locally ordered fluids bounded by planar interfaces. In the CDM theory, the geometry of the fluid region and the potential of mean force acting on the spin-bearing species determine the equilibrium distribution of spins as well as their translational dynamics. The latter is characterized by a time-dependent distribution function, or propagator, which, in the CDM theory, is obtained from the Smoluchowski mean-field diffusion equation, containing also the translational diffusivity tensor of the spin-bearing species. These distribution functions are then used to construct the time correlation functions and spectral densities that determine the observable spin relaxation rates. We showed in parts I and II that the predictions of the CDM theory may differ qualitatively from those of the traditional discrete

Brownian motion on a manifold

The question of the existence and correct form of equations describing Brownian motion on a manifold cannot be answered by mathematics alone, but requires a study of the underlying physics. As in classical mechanics, manifolds enter through the transformation of variables needed to account for the presence of constraints. The constraints are either due to a physical agency that forces the motion to remain on a manifold, or they represent conserved quantities of the equation of motion themselves. Also the Brownian motion is described either by a Smoluchowski diffusion equation or by a Kramers equation. The four cases lead to the following conclusions, (i) Smoluchowski diffusion with a conserved quantity reduces to a diffusion equation on the manifold; (ii) The same is true for diffusion with a physical constraint in three dimensions, but in more dimensions it may happen thatno autonomous equation on the manifold results; (iii) A Kramers equation with a conserved quantity reduces to an equation on the manifold, but in general not of the form of a Kramers equation; (iv) The Kramers equation with a physical constraint reduces to an autonomous Kramers equation on the manifold only for a special shape of that constraint. Throughout, only a certain type of physical constraints has been envisaged, and global questions are ignored. Finally, the customary heuristic construction of a Fokker-Planck equation for a mechanical system on a manifold is demonstrated for the case of Brownian rotation of a rigid body, and its shortcomings are emphasized.

Picosecond geminate charge recombination: the dependence of reaction kinetics on initial pair separation

The recombination kinetics of geminate electron-cation pairs in liquid hexane depends on the photoionization wavelength used for charge pair creation. Pairs generated by two-photon ionization of anthracene with 355 nm light exhibit a first half-life for recombination of 108 ps at 191 K; 266 nm light, 930 ps. Longer recombination times are attributed to a greater electron thermalization length in the liquid. The initial pair is studied both by the temperature dependence of the escape probability and by kinetic calculations based on the Smoluchowski diffusion equation.